A COUNTEREXAMPLE TO SEVERAL QUESTIONS ABOUT SCATTERED COMPACT SPACES RICHARD HAYDON 1. Introduction We give in this paper a negative answer to three questions that have been asked about the space #(AT) of continuous real-valued functions on a compact scattered space K. Two of the questions are about renormings and the other is about separately continuous functions and the Namioka Property. We recall that a Banach space X is said to be an Asplund space if, for every convex open subset E of X and every continuous convex function <J>: E -> IR, 0 is Frechet differentiable at all points of a dense ( S 6 subset of E. It is known that Zis such a space if and only if the dual space X* has the Radon-Nikodym Property and that this is so if and only if Y* is separable whenever Y is a separable subspace of A'[11,12,13]. Day [2, p. 167] raised the question of whether there exist implications between the Asplund Property and the existence of an equivalent norm that is (everywhere) Frechet differentiable. It is now known [7] that if X has a Frechet differentiable norm then X is an Asplund space, though the question of whether every Asplund space allows such a renorming has remained open. We shall exhibit a compact scattered space K such that %>(]£) (which is an Asplund space when AT is scattered [11]) admits not even a Gateaux differentiable renorming. We shall also show that the same ^(K) admits no strictly convex renorming. Again this is a rather stronger counterexample than might have been expected. For instance the problem is posed in [6] of whether the space #(AT) admits an equivalent norm that is locally uniformly convex (l.u.c.) whenever K is scattered. The first examples of Banach spaces not containing /^ and admitting no l.u.c. renorming appeared only recently [1, 9]. There is a closely related space which does not contain /^ and admits no strictly convex renorming; this will appear in [8]. A compact space K is said to have the Namioka Property (JV *) if, for every Baire space B and every separately continuous function <fi :B x K -*• U, there is a dense ^$ 8 subset H of B such that 0 is (jointly) continuous at all points of H x K. Equivalently, one may demand that every function tf/iB^^^K) which is continuous into the topology z p of pointwise convergence on K is continuous into the norm topology at all points of a dense ^S 6 . Talagrand [15] established that not every compact space has the Namioka Property and Deville [5] showed that the Stone-Cech compactification fiN does not have it. That the failure of the Namioka Property for K does not imply that ^(K) contains / OT was observed independently by Namioka (who considered the space constructed in [9]) and the present author (in [8]). In a positive direction, Debs [3] shows that all Corson compact spaces have the Namioka Property. Deville and Received 7 September 1989. 1980 Mathematics Subject Classification 46B20, 54D30. Bull. London Math. Soc. 22 (1990) 261-268